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Modeling of Energy States of Carriers in Quantum Dots

Modeling of Energy States of Carriers in Quantum Dots. Michael Yu. Petrov, St. Petersburg State University, Faculty of Physics e-mail: M.Yu.Petrov@gmail.com. OUTLOOK. Motivation Introduction into the Quantum Dot Heterostructures What is a quantum dot?

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Modeling of Energy States of Carriers in Quantum Dots

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  1. Modeling of Energy States of Carriersin Quantum Dots Michael Yu. Petrov, St. Petersburg State University, Faculty of Physics e-mail: M.Yu.Petrov@gmail.com

  2. OUTLOOK • Motivation • Introduction into the Quantum Dot Heterostructures • What is a quantum dot? • Self-organized semiconductor quantum dots • Energy Spectra • Modeling of Real Quantum Dots • Shape of real dots • Band profiles (including its modifications via strain effects) • Calculation models (effective mass approximation and multi-band k·p-method) • Optical transitions in real quantum dots (Coulomb interaction in excitons) • Applications of Modeling • Air-bridge detector device • Fock-Darwin spectra in ultra-high magnetic field • Optical transition of annealed quantum dots • Conclusion

  3. MOTIVATION • Quantum dot is a model object of fundamental research in modern semiconductor physics • Quantum dot is an object for applications and technology including: • Laser Technology • Optoelectronic Devices • Spintronics and Quantum Information Processing • Modeling because of a model object

  4. INTRODUCTIONINTO THE QUANTUM DOT HETEROSTRUCTURES WHAT IS A QUANTUM DOT? D. Bimberg, M. Grundmann, N.N. Ledentsov, Quantum Dot Heterostructures (Wiley, New York, 1999)

  5. SELF-ORGANIZED QUANTUM DOTS TEM of InAs/GaAs QDs (plan-view) V.G. Dubrovskii, G.E. Cirlin, et al., Journal of Crystal Growth 267 47-59 (2004). HRTEM of InP/InGaP QDs (front-view) Y. Masumoto, T. Takagahara, Semiconductor Quantum Dots: Physics, Spectroscopy and Applications, (Springer, Berlin, 2002).

  6. ENERGY SPECTRA(FROM BULK TO HETEROSTRUCTURES) Typical PL spectrum of InGaAs/GaAs QDs ExperimentallePhysik II, Universitaet Dortmund, Germany D.Bimberg, M.Grundmann, N.N.Ledentsov, Quantum Dot Heterostructures (Wiley, New York, 1999)

  7. SIMPLEST MODELS OF ENERGY STRUCTURE • Cube-like QD with infinite barriers • Sphere-like QD with infinite barriers For InAs QD (me=0.023m0): cube: a=10 nm E111=0.49 eV sphere: R0=6.2 nm E10 =0.42 eV (cube volume = sphere volume)

  8. MODELING OF REAL QUANTUM DOTS • Important parameters for real QDs: • shape and volume of QDs in sample • band profiles (including its modification via strain) • Different methods of calculation of energy structure: • one-band effective mass approximation • multi-band calculations • Coulomb interaction of carriers

  9. SHAPE AND VOLUME OF QUANTUM DOTS • A “regularly shaped” QDs are available only atexcellent growthconditions • Size spread is approximately 10% for self-organized QDs • It is not possible to describe the QD ensemble by microscopy of single QD • Two most popular models of QD shape: pyramid, lens

  10. STRAIN PROFILES IN QUANTUM DOTS • Harmonic Continuum Elasticity Theory (CE) • Atomistic Valence-Force-Field Model (VFF) The solution for strain tensor, εij, can be obtain by minimizing the elastic energy, ECE, by modifying the displacement vectors, ui The solution for strain tensor, εij, can be obtain by minimizing the elastic energy, ECE, by modifying the atomic positions

  11. STRAIN PROFILES IN QUANTUM DOTS(CONTINUE) C. Pryor et al., J. Appl. Phys. 83, 2548-2554 (1998)

  12. INFLUENCE OF STRAIN ON BAND PROFILES C. Pryor, Phys. Rev. B 57, 7190-7195 (1998)

  13. COMPARISON OF DIFFERENT METHODS OF CALCULATION OF ENERGY STATES OF CARRIERS C. Pryor, Phys. Rev. B 57, 7190-7195 (1998)

  14. ELECTRON AND HOLE DENSITIES O. Stier, M. Grundmann, D. Bimberg, Phys. Rev. B 59, 5688-5701 (1999)

  15. OPTICAL EXCITONIC TRANSITIONS • Strong Confinement Regime (simpleconsideration) • Hartree Approximation Ee E= Ee + Eh -EX O. Stier, M. Grundmann, D. Bimberg, Phys. Rev. B 59, 5688-5701 (1999) Eh

  16. EXCITONIC SPECTRUM OF INGAAS QUANTUM DOTS 1e-1h 2e-2h 3e-3h

  17. MODIFICATIONS OF THE ELECTRONIC STATES OF InGaAs QUANTUM DOTS EMBEDDED IN BOWED AIRBRIDGE STRUCTURES left-up: SEM of structure; right: PL spectrum; left-down: Energy Shift T. Nakaoka, T. Kakitsuka, et al., Journ. Appl. Phys. 94, 6812 (2003).

  18. INFLUENCE OF ULTRA-HIGH MAGNETIC FIELD ON ENERGY STRUCTUREOF InGaAs/GaAs QUANTUM DOTS Fock-Darwin spectrum (left (c) – experiment, right – 8-band k·p-model) S. Raymond, S. Studenikin, et al., Phys. Rev. Lett. 92, 187402 (2004).

  19. MODELING OF ENERGY SPECTRA OF ANNEALEDINAS/GAAS QUANTUM DOTS • Bell-like shaped QD for describing the average in ensemble • Diffusion Law for describing thermal annealing • Model of Constant Potentials for carriers • One-band Effective Mass Approximation for energy states calculations z M.Yu. Petrov, I.V. Ignatiev et al., Phys. Rev. B (submitted); also available in arXiv: 0710.5091v4

  20. INTERDIFFUSION OF INDIUM AND GALLIUMDUE TO THERMAL ANNEALING OF QUANTUM DOTS EA a

  21. MODIFICATION OF CARRIER DENSITIES DUE TO THERMAL ANNEALING Electron density distribution Indium concentration distribution Hole density distribution

  22. EXCITONIC SPECTRA OF ANNEALED QUANTUM DOTS

  23. CONCLUSION • The basic principles of calculations of energy structure of quantum dots were demonstrated • The main important parameter is a built-in strain • For approximation of lowest state the simplest constant potential models of QD can be used • Describing of excited states requires more complex models (band mixing, coulomb interaction etc.)

  24. Thank You For Your Attention!

  25. REFERENCES • D. Bimberg, M. Grundmann, N.N. Ledentsov, Quantum Dot Heterostructures (Wiley, New York, 1999). • Y. Masumoto, T. Takagahara, Semiconductor Quantum Dots: Physics, Spectroscopy and Applications, (Springer, Berlin, 2002). • C. Pryor et al., J. Appl. Phys. 83, 2548-2554 (1998). • C. Pryor, Phys. Rev. B 57, 7190-7195 (1998). • O. Stier, M. Grundmann, D. Bimberg, Phys. Rev. B 59, 5688-5701 (1999). • T. Nakaoka, T. Kakitsuka, et al., J. Appl. Phys. 94, 6812 (2003). • S. Raymond, S. Studenikin, et al., Phys. Rev. Lett. 92, 187402 (2004). • M.Yu. Petrov, I.V. Ignatiev, et al., Phys. Rev. B (submitted); also available in arXiv: 0710.5091v4

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